Hi, i was just wondering if you could help me with how to work out this question, as i have attempted and get a bit confused at the end. here it is:
A cylinder is inside a sphere- the sphere has a radius of R. What is the maximum volume of the cylinder?
Thanks - really appreciated :)
make a sketch
let the radius of the cylinder be r, let the height of the cylinder be 2h , (avoiding fractions that way)
then r^2 + h^2 = R^2 , ---> r^2 = R^2 - h^2
vol of cyl = πr^2(2h)
= 2π(R^2 - h^2)h
= 2π(hR^2 - h^3)
d(vol)/dh = 2π(R^2 - 3h^2) = 0 for a max of volume
3h^2 = R^2
√3 h = R
h = R/√3
then r^2 = R^2 - R^2/3 = (2/3)R^2
max volume = πr^2(2h)
= π(2R^2/3)(2R/√3)
= (4/3)πR^3 (1/√3)
They might want a rationalized answer.
I purposely left it like that to point out that the first part of the answer is simply the volume of the sphere with radius R.
So in general, the maximum volume of the cylinder is
1/√3 of the volume of the sphere that contains it.
Of course, I'd be happy to help you with that question! To find the maximum volume of the cylinder inside a sphere, we can break down the problem into smaller steps.
Step 1: Understand the problem
Let's visualize the problem. We have a sphere with radius R, and we want to find the maximum volume of a cylinder that can fit inside the sphere. To find this volume, we need to determine the radius and height of the cylinder.
Step 2: Identify the key components
In this case, the key components are the sphere and the cylinder. We need to find the relationship between the sphere's radius (R) and the cylinder's radius and height.
Step 3: Determine the relationship between the sphere and the cylinder
Since the cylinder is inscribed inside the sphere, the sphere's diameter will be equal to the cylinder's height. This is because the diameter of the sphere is twice the radius, while the height of the cylinder is the same as its diameter. So, the height of the cylinder is 2R.
Step 4: Find the radius of the cylinder
To find the radius of the cylinder, we need to consider that the cylinder's base is a circle and is inscribed inside the sphere. This means the circle's diameter (which is equal to the cylinder's diameter and twice its radius) is equal to the sphere's diameter (which is twice its radius). So, the diameter of the cylinder's base is also 2R, which means the radius of the cylinder is R.
Step 5: Calculate the maximum volume of the cylinder
The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. Now that we know the radius (R) and height (2R) of the cylinder, we can substitute these values into the formula:
V = π(R)²(2R)
= 2πR³
So, the maximum volume of the cylinder that can fit inside the sphere is 2πR³.
I hope this explanation helps you understand how to approach and solve this problem. Let me know if you have any further questions!